One logical law that is easy to
accept is the law of non-contradiction. This law can be expressed by the
propositional formula ¬(p^¬p). Breaking the sentence down a little
makes it easier to understand. p^¬p means that p is both true and
false, which is a contradiction. So, negating this statement means that
there can be no contradictions (hence, the name of the law). In other
words, the law of non-contradiction tells us that a statement cannot be
both true and false at the same time. This law is relatively uncontroversial,
though there have been those who believe that it may fail in certain special
cases. However, it does lead us to a logical principle that has historically
been more controversial: the law of excluded middle.
The law of excluded middle can be
expressed by the propositional formula p_¬p. It means that a statement
is either true or false. Think of it as claiming that there is no middle
ground between being true and being false. Every statement has to be one
or the other. That’s why it’s called the law of excluded middle,
because it excludes a middle ground between truth and falsity. So while
the law of non-contradiction tells us that no statement can be both true
and false, the law of excluded middle tells us that they must all be one
or the other. Now, we can get to this law by considering what it means
for the law of non-contradiction to be true. For the law of noncontradiction
to be true, ¬(p^¬p) must be true. This means p^¬p must be
false. Now, we must refer back the truth table definition for a conjunction.
What does it take for p ^ ¬p to be false? It means that at least one
of the conjuncts must be false. So, either p is false, or ¬p is false.
Well, if p is false, then ¬p must be true. And if ¬p is false,
then p must be true. So we are left with the disjunction p _ ¬p, which
is exactly the formulation I gave of the law of excluded middle. So we
have just derived the law of excluded middle from the law of non-contradiction.
What we just did was convert the
negation of a conjunction into a disjunction,
by considering what it means for the conjunction to fail. The rule that
lets us do this is known as De Morgan’s rule, after Augustus De
Morgan. Formally speaking, it tells us that statements of the following
two forms are equivalent: ¬(p^q) and ¬p_¬q. If you substitute
¬p for q in the first formula, you will have the law of non-contradiction.
So you might want to try doing the derivation yourself. You will, however,
need the rule that tells us that p is equivalent to ¬¬p. The point
of this exercise, however, was to show that it is possible to derive the
law of excluded middle from the law of non-contradiction. However, it
is also possible to convince yourself of the truth of the law of excluded
middle without the law of non-contradiction.
We can show using the method of truth
tables that the disjunctive statement
p _ ¬p is always true. As a point of terminology, a statement that
is always true, irrespective of the truth values of its components, is
called a tautology. p _ ¬p is a tautology, since no matter what the
truth value of p is, p_¬p is always true. We can see this illustrated
in the truth table below.

p ¬p p _ ¬p
T F T
F T T

You can try constructing a similar
truth table to show that the law of non-contradiction is also a tautology.
Its truth table is a bit more complicated, though. However, since the
law of excluded middle is a tautology, it should hold no matter what the
truth value of p is. In fact, it should be true no matter what statement
we decide p should represent. So the law of excluded middle tells us that
every statement whatsoever must be either true or false. At first, this
might not seem like a very problematic claim. But before getting too comfortable
with this idea, we might want to consider Bertrand Russell’s famous
example: “The present King of France is bald.” Since the law
of excluded middle tells us that every statement is either true or false,
the sentence “The present King of France is bald” must be
either true or false. Which is it?
Since there is no present King of
France, it would seem quite unusual to claim that this sentence is true.
But if we accept the law of excluded middle, this leaves us only one option
- namely, to claim that it is false. Now, at this point, we might choose
to reject the law of excluded middle altogether, or contend that it simply
does not hold in some cases. This is an interesting option to consider,
but then we might need to consider why the method of constructing truth
tables tells us that the law of excluded middle holds, if it actually
doesn’t. We would also have to consider why it is derivable from
the principle of non-contradiction. After all, this sentence doesn’t
pose a problem for the law of non-contradiction, since it’s not
both true and false. So we’ll ignore this option for now.
Returning to the problem at hand,
we must now consider the question of what it means for the sentence “The
present King of France is bald” to be false. Perhaps it means “The
present King of France is not bald.” But that would imply that there
is a present King of France, and he is not bald. This is not a conclusion
we want to reach. Russell, in his 1905 paper “On Denoting”
presented his own solution to this problem, which comes in the form of
a theory of definite descriptions. Under this theory, we can think of
there being a hidden conjunctive structure in the sentence “The
present King of France is bald.” What the sentence really says is
that there is a present King of France, and he is bald. So the fact that
there is no present King of France implies that this sentence is false,
and we have the solution we need.
Russell’s solution clearly
suggests that we can’t just extract the logical structure of a sentence
from its grammatical structure. We have to take other things into account.
If you’re interested in issues about the relationship between logic
and language, you might want to take a class in philosophy of language.
The other essay in this section, entitled “Logic and Natural Language”,
covers some other issues in this area.
One method of proof that comes naturally
from the law of excluded middle is a proof by contradiction, or reductio
ad absurdum. In a proof by contradiction, we assume the negation of a
statement and proceed to prove that the assumption leads us to a contradiction.
A reductio ad absurdum sometimes shows that the assumption leads to an
absurd conclusion, which is not necessarily a contradiction. In both cases,
the unsatisfactory result of negating our statement leads us to conclude
that our statement is, in fact, true. How does this follow from the law
of excluded middle? The law of excluded middle tells us that there are
only two possibilities with respect to a statement p. Either p is true,
or ¬p is true. In showing that the assumption of ¬p leads us to
a contradictory conclusion, we eliminate the possibility that ¬p is
true. So we are then forced to conclude that p is true, since the law
of excluded middle is supposed to hold for any statement whatsoever.
Now, I’ve been a bit flippant
in talking about statements. Statements can be about a lot of different
things. The above discussion illustrated a problem with statements about
things that don’t actually exist. I’m sure most people would
agree that the designation “the present King of France” refers
to something that doesn’t exist. But what about situations where
it’s not so certain? One of the main metaphysical questions in the
philosophy of mathematics is the question of whether or not mathematical
objects actually exist. Think about the question of whether numbers exist.
If they do exist, then what are they? After all, they’re not concrete
things that we can reach out and touch. But if they don’t exist,
then what’s going on in math?
Metaphysical worries have motivated
certain people to argue that proofs by contradiction are not legitimate
proofs in mathematics. Proponents of intuitionism and constructivism in
mathematics place a significant emphasis on the construction of mathematical
objects. One way to characterize this position is that in order to show
that a mathematical object exists, it is necessary to construct it, or
at the very least, provide a method for its construction. This is their
answer to the metaphysical question. Suppose we had a mathematical proof
in which we assumed an object did not exist, and proved that our assumption
lead us to a contradiction. For an intuitionist or a constructivist, this
proof would not be a sufficient demonstration that the object does exist.
A sufficient demonstration would have to involve the construction of the
object.
Even if questions about existence
get too complicated, we can still ask the question “What mathematical
objects can we legitimately talk about?” The intuitionist answer
is that we can talk about those mathematical objects which we know can
be constructed.
Simply speaking, intuitionistic logic
is logic without the law of excluded middle. I have outlined some small
part of the motivation behind developing such a system, but more details
can be found in the work of L.E.J. Brouwer and Arend Heyting.